As of the Dashboard update that came out during the week of December 5th, you can set Beacons on up to 5 games that let another user know that even if you're not playing it, you like playing a particular game, and may not mind being invited to play.

When exactly do these Beacons fire (from both things you do, and things other people do), and who all will the event affect?

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When you set up a beacon you can initially choose to just share it with your xbox live friends but also on facebook. After sharing, your friends will be able to see your beacon in different ways. One of them is when they launch the game you want to play, then they will get a notification saying "Gamertag wants to play this game". The other way is for them to go to the beacons section where they can see a list of all their friends' beacons.
You will get a notification everytime one of your friends launches a game that you have set up a beacon for as well so you can contact them.